A Comparative Analysis of Alternative Maximum Probable Yearly Aggregate Loss Estimators

Abstract

This article discusses methods for estimating the maximum probable yearly aggregate loss (MPY), a fractile point in the right tail of the distribution of total annual loss costs. The best method for determining MPY is to estimate the loss cost distribution either analytically or by simulation. Approximation formulas also are available. These are analyzed and their estimates of MPY are compared with simulation estimates. The results suggest that the normal power method is an accurate approximation technique and that the simulation results are extremely sensitive to the choice of distribution. The non-central t distribution is suggested to represent severity. The traditional focus of risk management on loss identification, reduction, and transfer gradually is being supplanted by a more balanced approach, with retention playing a prominent role. This evolutionary process has directed attention to techniques for making decisions about retention plans. Such decisions should be made by evaluating a company's loss experience according to theoretically sound business decision rules. While various decision rules have been suggested, expected utility decision making is considered by many to be the best method. Although both the theory and the technology are available to implement utility theory principles,' such methods probably will not be applied on a widespread basis for a number J. David Cummins, Ph.D., CLU, is Associate Professor of Insurance in the Wharton School of the University of Pennsylvania. He is author of An Econometric Model of the Life Insurance Sector of the U.S. Economy, editor of Investment Activities of Life Insurance Companies, co-author of Impact of Consumer Services on Independent Insurance Agency Performance, and a member of the editorial board of the Journal of Risk and Insurance. Leonard R. Freifelder, Ph.D., is Assistant Professor in Operations Research in the School of Business, Temple University. He is author of a monograph-A Decision Theoretic Approach to Insurance Ratemaking (1976); a statistical consultant for Professional Market Research (1973-1976); and is an Associate Fellow of the Society of Actuaries. The authors are grateful to the Risk Studies Foundation for providing financial support for the research reported in this article. They also thank Frederick N. Nowell III, Risk Management Department, Brewer & Lord, Boston, Mass. for his helpful suggestions and cooperation. Valuable comments on earlier drafts of the article were provided by Robert Goodrich and by two anonymous referees. Of couse, any errors in the article are the sole responsibility of the authors. ' A theoretical approach to retention decisions is developed in Cummins (7), while an application of decision theory to a risk management problem is discussed in Shpilberg and de Neufville (26).